Asymptotics for cliques in scale-free random graphs

TL;DR

This paper derives detailed asymptotic formulas for the expected number of cliques of all sizes in a scale-free inhomogeneous random graph model, highlighting differences between integer and non-integer degree exponents.

Contribution

It provides a comprehensive set of asymptotic results for clique counts in the Chung--Lu model with power-law weights, including non-integer exponents and partial results for integer exponents.

Findings

Complete asymptotics for all clique sizes with non-integer alpha

Explanation of differences in asymptotics for integer alpha

Partial results for integer alpha case

Abstract

In this paper we establish asymptotics (as the size of the graph grows to infinity) for the expected number of cliques in the Chung--Lu inhomogeneous random graph model in which vertices are assigned independent weights which have tail probabilities $h^{1-\alpha}l(h)$, where $\alpha>2$ and $l$ is a slowly varying function. Each pair of vertices is connected by an edge with a probability proportional to the product of the weights of those vertices. We present a complete set of asymptotics for all clique sizes and for all non-integer $\alpha > 2$. We also explain why the case of an integer $\alpha$ is different, and present partial results for the asymptotics in that case.